Mathematics for the interested outsider

Simple and Elementary Functions

We now introduce two classes of functions that are very easy to work with. As usual, we’re working in some measurable space.

First, we have the “simple functions”. Such a function is described by picking a finite number of pairwise disjoint measurable sets and a corresponding set of finite real numbers . We use these to define a function by declaring if , and if is in none of the . The very simplet example is the characteristic function of a measurable function . Any other simple function can be written as

Any simple function is measurable, for the preimage is the union of all the corresponding to those , and is thus measurable.

It’s straightforward to verify that the product and sum of any two simple functions is itself a simple function — given functions and , we have and . It’s even easier to see that any scalar multiple of a simple function is simple — . And thus the collection of simple functions forms a subalgebra of the algebra of measurable functions.

“Elementary functions” are similar to simple functions. We slightly relax the conditions by allowing a countably infinite number of measurable sets and corresponding values .

Now, why do we care about simple functions? As it happens, every measurable function can be approximated by simple functions! That is, given any measurable function we can find a sequence of simple functions converging pointwise to .

To see this, first break up into its positive and negative parts and . If we can approximate any nonnegative measurable function by a pointwise-increasing sequence of nonnegative simple functions, then we can approximate each of and , and the difference of these series approximates . So, without loss of generality, we will assume that is nonnegative.

Okay, so here’s how we’ll define the simple functions :

That is, to define we chop up the nonnegative real numbers into chunks of width , and within each of these slices we round values of down to the lower endpoint. If , we round all the way down to . There can only ever be values for , and each of these corresponds to a measurable set. The value corresponds to the set

while the value corresponds to the set . And thus is indeed a simple function.

So, does the sequence converge pointwise to ? Well, if , then for all . On the other hand, if then ; after this point, and are both within a slice of width , and so . And so given a large enough we can bring within any desired bound of . Thus the sequence increases pointwise to the function .

But that’s not all! If is bounded above by some integer , the sequence converges uniformly to . Indeed, once we get to , we cannot have for any . That is, for sufficiently large we always have . Given an we pick an so that both and , and this will guarantee for every. That is: the convergence is uniform.

This is also where elementary functions come in handy. If we’re allowed to use a countably infinite number of values, we can get uniform convergence without having to ask that be bounded. Indeed, instead of defining for , just chop up all positive values into slices of width . There are only a countably infinite number of such slices, and so the resulting function is elementary, if not quite simple.

Share this:

Like this:

Related

[…] We start our turn from measure in the abstract to applying it to integration, and we start with simple functions. In fact, we start a bit further back than that even; the simple functions are exactly the finite […]

[…] before, but now for general integrable functions. Similarly, if is nonnegative a.e., we can find a sequence of nonnegative simple functions converging a.e. (and thus in measure) to . The integral […]

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.